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Fractional Brownian Motions in a Limit of Turbulent TransportAuthor(s): Albert Fannjiang and Tomasz KomorowskiSource: The Annals of Applied Probability, Vol. 10, No. 4 (Nov., 2000), pp. 1100-1120Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/2667220Accessed: 14/05/2010 14:32

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The At?nnals of Applied Probability 2000, Vol. 10, No. 4, 1100-1120

FRACTIONAL BROWNIAN MOTIONS IN A LIMIT OF TURBULENT TRANSPORT1

BY ALBERT FANNJIANG AND TOMASZ KoMOROWSKI

University of California, Davis and Maria Curie-Sktodowska University, and Polish Academy of Sciences

We show that the motion of a particle advected by a random Gaussian velocity field with long-range correlations converges to a fractional Brownian motion in the long time limit.

1. Introduction. The motion of a particle advected by a random velocity field is governed by

(1) dx(t) = V(t, x(t)), dt

where V(t, x) = (Vl(t, x), Vd(t, x)) is a random, mean-zero, time- sta- tionary, space-homogeneous incompressible velocity field in dimension d > 2.

In certain situations, it is believed that the convergence of the Taylor-Kubo formula (see [8] and [14]) given by

00

(2) f{E[Vi(t, O)Vj(O, 0)] + E[Vj(t, O)Vi(O, O)]} dt

is a criterion for convergence of turbulent motion to Brownian motion in the long time limit. Indeed, it has been shown that the solution of

(3) dt !V( , x,(t)), x8(0) = 0,

converges in law, as e -> 0, to the Brownian motion with diffusion coefficients given by the Taylor-Kubo formula when the velocity field is sufficiently mixing in time (see [2], [6], [7] and [9]). Moreover, the solution of (3) converges to the same Brownian motion for a family of nonmixing Gaussian, Markovian flows with power-law spectra as long as the Taylor-Kubo formula converges (see [3]). In this paper, for the same family of power-law spectra, we show that, when the Taylor-Kubo formula diverges, the solution of the following equation

(4) dx,(t) _1-28V( c2v x8(t))v x8(0) = O,

Received June 1999; revised December 1999. 1The research of Fannjiang was supported in part by NSF Grant DMS-96-00119. This work

was finished while Komorowski was visiting the Department of Statistics, University of California, Berkeley.

AMS 1991 subject classifications. Primary 60F05,76F05,76R05; secondary 58F25. Key words and phrases. Turbulent diffusion, mixing, fractional Brownian motion.

1100

FRACTIONAL BROWNIAN MOTION LIMIT 1101

with some 8 -7 1 depending on the velocity spectrum, converges, as 8 -> 0, to a fractional Brownian motion (FBM), as introduced in [10] (see also [13]).

We define the family of velocity fields with power-law spectra as follows. Let (fQ, X/, P) be a probability space of which each element is a velocity field V(t, x), (t, x) E R x Rd, satisfying the following properties:

(HI) V(t, x) is time stationary, space homogeneous, centered, that is, E{V} = 0, and Gaussian. Here E stands for the expectation with respect to the probability measure P.

(H2) The two-point correlation tensor R = [Rij] is given by

R jj(t, x) E [Vi (t, x) Vj (, O)]

(5)

- f

cos (k

.

x)elk12ltRij(k dk

with the spatial spectral density

(6) R(k) = ak D(a+d)2 kI- ok)'

where a: [O, +oc) -- R+ is a compactly supported, continuous, nonneg- ative function. The factor I - k 0 k/JkJ2 in (6) is a result of incompress- ibility.

(H3) a < 1, /8 > 0 and a +,8 > 1.

It can be readily checked that the correlation function (5) is temporally integrable and, hence, the Taylor-Kubo formula is convergent if and only if a +/3 < 1.

The function exp (- Ik12't) in (5) is called the time correlation function of the flow V. For /8 > 0, the velocity field lacks the spectral gap and, thus, is not mixing in time. As the time correlation function is exponential, the Gaussian velocity field is an Ornstein-Uhlenbeck process. Because the function a has a compact support we may assume, without loss of generality, that V is jointly continuous in both (t, x) and is C" in x almost surely. For a < 1, the spectral density R(k) is integrable in k and, thus, (5)-(6) defines a random velocity field with a finite second moment. The exponent a is directly related to the decay exponent of R. Namely, JRJ(0, x) - Ixal- for lxl >> 1. As a increases to 1, the decay exponent of R decreases to 0.

Our main result is summarized in the following theorem (see also Figure 1).

THEOREM 1. Under assumptions (H1)-(H3), the solution of (4) with the scaling exponent

(7) /3 a+ 2,8 - I

converges in law, as ? tends to 0, to a fractional Brownian motion BH(t), that is, to a Gaussian process with stationary increments whose covariance is given by

(8) E[BH(t) 0 BH(t)] = Dt2H,

1102 A. FANNJIANG AND T. KOMOROWSKI

\B o

\Fractipnal Brownianl Motion Limit

Brownian Motion Limit

a=O a=1

FIG. 1. Phase diagram for scaling limit.

with the coefficients D

(9) D = | ek 1 +|k|P( kI k) a(O) dk Id Ik 12a+4/3 l Jk 2 ) k d-1

and the Hurst exponent H

(10) < H= - + < 1. 2 2 2,3

Moreover, we show that the process x?(t) is asymptotically, as ? - 0, the same as the process

t/? 28

Y8(t) X?(0) + f V(s, x?(0)) ds

(see Section 4). Namely, the spatial dependence of the Lagrangian velocity is frozen. As a result, the asymptotic motions of N particles starting at xlj(0), x2 (0),.. , xN(O), can be easily deduced and they are in stark contrast to the case of diffusive scaling (cf. [2], [3]).

REMARK. Molecular diffusion can be added to the equation of motion so that instead of (1) we may consider an Ito stochastic differential equation

dx(t) = V(t, x(t)) dt + 2K dB(t),

with B(t), t > 0, the standard Brownian motion, independent of V and K > 0.

This, however, would not influence our results.


2. Multiple stochastic integrals. By the spectral theorem (see, e.g., [1]) we assume without loss of any generality that there exist two independent, identically distributed, real vector-valued, Gaussian spectral measures VI(t, ), I = 0, 1, such that

(11) V(t, x) = fVo(t, x, dk),

where

Vo(t, x, dk) co (k x)Vo(t, dk) + c1(k x)Vl(t, dk), with co(4) cos (4), cl(+) -sin (4). Define also

V1(t, x, dk) -c1(k x)Vo(t, dk) + co(k x)Vl(t, dk). We have the relations

(12) dVo(t, x, dk)/dxj = kjV,(t, x, dk),

(13) dV,(t, x, dk)/dxj = -kiVo(t, x, dk).

Clearly, fV1(t, x, dk) is a random field distributed identically to and indepen- dently of V. We define the multiple stochastic integral

(14) ... |4(kl,.. ., kN)Vll(tl, xl, dkl) ... ( VIN(tN, xN, dkN)

for any 1, . IN E {0, 1} and a suitable family of functions Ef by using the Fubini theorem [see (15)]. For qf1, 7/N E Y(Rd), the Schwartz space, and 11, ,IN E {O, 1} we set

| '|l(kl) ..

' 'N(kN)Vll(tl, Xl, dkl) XD ..X VIN (tN, XN, dkN)

(15)

:= |j(kj)VI1(tj, Xl, dkl) X N (kN)VIN (tN, XN, dkN)-

We then extend the definition of multiple integration to the closure X of the Schwartz space ((Rd)N, R) under the norm

11X112:f-|.f r(klk ,kN)q(kl, ...,kN)

(16) x E[Vl,(tl,xl,dkl)?(... (VIN(tN,XN,dkN)

VI Vl(tl,xl, dk/)g 0 VIN (tN, XN, dk'N]

The expectation is to be calculated by the formal rule

E [VI, i (t, x, dk)V,,, j, (t', x', dk')]

= eHk2 tt'l8 ,co(k. (x - x'))Ri , (k)8(k - k') dk dk'.

This approach to spectral integration follows [12]. When i = (il, ., id), il, ., id E {1, 2,..., d}, is fixed and l = (11, ...,IN),

11, . . ., IN E {0, 1}, we shall denote the corresponding component of the stochastic integral by M, i


Note that ', i E HN(V)-the Hilbert space obtained as a completion of the

space of Nth-degree polynomials in variables f qf(k)V(t, x, k) with respect to the standard L2-norm.

PROPOSITION 1. For any (tl, X1), ..., (tN, XN) E R x Rd and p > 0, 1 i belongs to LP(fl) and

(17) (EvIr1,iIP)l/P < C(ELIr1,i12)1/2,

with the constant C depending only on p, N and the dimension d. Moreover, NV i is differentiable in the mean square sense with

vxj Il, i (t 1 t *-- X1, *-- -,XN)

(18) .. k-)j | ky,/(k,..., kN) V11i1,e(t1, x1,dk1) ..Vl-l j, ii (t, vxji dkj)

* VN, iN(tN,XN, dkN)

The proof of Proposition 1 is standard and follows directly from the well- known hypercontractivity property for Gaussian measures (see, e.g., [5], Theorem 5.1 and its corollaries), so we do not repeat it here.

The field V is Markovian, that is,

E[f q(k)V1(t, x, dk) x, s (19)

e fe-kl2f(ts)qJ(k)V1(s, x, dk), 1 = 0, 1,

for all qi E J(Rd, R), where 7 b denotes the ou-algebra generated by random variables V(t, x), for t E [a, b] and x E Rd.

To calculate a mathematical expectation of multiple products of Gaussian random variables, it is convenient to use a graphical representation, borrowed from quantum field theory. We refer to, for example, Glimm and Jaffe [4] and Janson [5]. A Feynman diagram Y (of order n > 0 and rank r > 0) is a graph consisting of a set B(Y) of n vertices and a set E(y) of r edges without common endpoints. So there are r pairs of vertices, each joined by an edge, and n - 2r unpaired vertices, called free vertices. Note that B(S) is a set of positive integers. An edge whose endpoints are m, n E B is represented by mTn (unless otherwise specified, we always assume m < n); an edge includes its endpoints. A diagram Y is said to be based on B(S). Denote the set of free vertices by A(y), so A(y) = Y \ E(y). The diagram is complete if A(F) is empty and incomplete, otherwise. Denote by S(B) the set of all diagrams based on B, by S4(B) the set of all complete diagrams based on B and by Si(B) the set of all incomplete diagrams based on B. A diagram 7' E 4,(B) is called a completion of YS E Si (B) if E(F) c E(7').

Let B = {1, 2, 3, ..., n}. Denote by 7k the subdiagram of $, based on {1, ..., 14. Define Ak(Q9) = A(Q5k). A special class of diagrams, denoted by Ss(B), plays an important role in the subsequent analysis: a diagram 7 of


order n belongs to 9s(B) if Ak(y) is not empty for all k = 1, ..., n. We shall adopt the following multiindex notation. For any P E Z+, multiindex n = (n1, ..., np), n1 stands for E ni. If P' < P we denote nlp, :(nl, .. ., np,). In addition, if k is any number we set n* k : (nl, ..., np, k). We work out the conditional expectation for multiple spectral integrals using the Markov property (19).

PROPOSITION 2. For any function i E X and I1., IN E {0, 1}, ii,.... iN E {1 d,d}

E q -f(kl~ ... kN) VI,, il (t, xl dkl) --VIN, iN (t" N dkN) 7/~-oo s]

(20) = .N . exp - rn n7 1kn2/(t_s)} ,7ES{1---NJ) MEA(,7)

x qf(kl,..., kN)V S, Xl ..,XN (dkl, ...,dkN; 1),

with

V S, Xl, ... ,XN (dkl, . , dkN; 17)

(21) H V1 , i7Z (S, Xm, dknZ) f [I -e(k 12/+1k12/) (t-s)] zneA(S9) iaD1EE(7)

x E[V,, il (s, xm, dkm) Vl, Z (s, xii , dki,)]

PROOF. Without loss of generality we consider q(kl, .. ., kN) = lA1(kl)...

lAN(kN) for some Borel sets A1, ..., AN. Note that VI(t, Ai) = VI (t, Ai) +

V1(t, Ai), where V? (t, -) is the orthogonal projection of Vj(t, .) on L and

Vi (t, ) its complement. Here La b denotes L2 closure of the linear span over V(s, x), a < s < b, x E Rd. The conditional expectation in (20) equals

H r| E[VLi",i (t, Ain)Vz 1, 1(t, An V) (t, Am). YES({1,...,N}) f-nEE(Y) mEA(Y7)

The statement follows upon the application of the relations

V?(t, A) = 'A e-1k12(t-S) VI(s dk)

and

I(t, A) JR [(t, B)k

= || ,1X |E[i(t, dk)OVI,(t, dk')]-E[VI?t d)VI?,(t k)


3. Proof of tightness. Let us start with the following result, which estab- lishes, among other things, that the family of continuous trajectory processes x?(t), t > 0, is tight.

LEMMA 1. For the family of trajectories given by (4) we have

lim E[(x(t) - x(?(u)) 0 (x?(t) - x((u))] D(t - u)2H if t > u, 4~0

where H and D are given by (9) and (10), respectively.

PROOF. First, let us observe that since x?(t) has stationary increments it is enough to prove the lemma for u = 0. By the stationarity of V(s, ?x(s)) (see [11]), we can write that

(22) lim E[x8 (t) 0 xl:(t)] = lim g2 f ds E[V(s', cx(s')) ? V(0, 0)] ds'. 4~0 4~0 0

Thus (22) equals N

(23) 2 E >7 + N, z=1

where

,t= 7z+10 dsj ds, j E[W7 l(si, Sn, 0) 0V(O, 0)] ds,

and

Wo(s1, x) = V(s1, x),

Wil(sl, ,~sn+l? x) = V(s12i, x) .VWn-1(sl, ... ,sxfx) fornn1,2, ...

with the remainder term

(24 =26NN+2 ds dsl... E[WN(S1, ..SN+. 1 X(SN+1))

?V(0, 0)] dSN+l'

Estimates of f7. Elementary calculations show that

limY- 1 = Dt2H. 4t0

Since V is Gaussian we deduce that

E,e' = 0,

when n > 2 is even. We now show that

(25) limEY7 0 4~0

for n > 3 odd. The i, jth entry of the matrix 1n is given by

(6 J - 2671+11 dsf ds, .| E[EoVVz, r ( . s70)

x Vj(O, 0)] ds7,.


We would like to express the conditional expectation appearing in (26) in terms of spectral measures associated with the velocity field. To do so, we introduce first the so-called proper functions of order n, a- {1, ..., n} -> {0, 1} that appear in the statement of the next lemma. The proper function of order 1 is unique and is given by o-(1) = 0. Any proper function, o-', of order n + 1 is generated from a proper function a- of order n as follows. For some p < n,

u'(n + 1) O,

(27) u-'(k) ov(k) for k < n and k A p,

uJ'(p) 1 - o-(p).

In other words, each proper function o- of order n generates n different proper functions of order n + 1. Thus, the total number of proper functions of order n is (n - 1)!. In the remainder of the paper, we sometimes write ok instead of or(k).

LEMMA 2. Let n > 1 and s, i S2 > * 7 7,i E{, .. , dl, x E- R.

We have then that

ES1i+1 W n-1, Ji(.s , s1l, x)

:=E[W11_1, Ji(Sl. * Sn n x)l )Noo S'2+1 I

(28) = > 5if. .fc)(k1, , k17) exp j- L _km l (s - s1+?)}

x Pn_1(Y)Q(Y) Hl Vin, 1 x(sn+IX, dk7),

MEA,, (n)

where q7z) are some functions, with sup l< 1, fl

~~ ~ km Iexp ~

12 2Sj_ S+j (29) Pnz_1(Y) =l Y1 ( kl ePt Ik77lps-yl} (j=1 'MEA j(7) mEAj(7)

(30) Q(Y5) = H E[Vi,l,J,,JZ,O, dkm)Vi,,,,,rl,o (O, dkin')]. In7in' EE(97)

The summation is over all multiindices i of length n, whose first component equals i, all Y cEA and all proper functions u- of order n.

Before proving Lemma 2, we apply it to show (25). Notice that according to (28),

10 ds dsi .. EOWn-1,i(Sl Sn, 0)ds,

= Li dsJ J.o(. )(kj ...,k1) eXPj- L k12,}

X P7z_1(Y)Q(1_) fl Vj,,ff,l(mAdkm() MEA,j (7)


for i = 1, ... , d. Here, adopting the convention sn+1 := 0, we set

71i (kl,..., kj

JJd~... fsnl-1 4~.k.,)xH~ x{ mA(7 ,~I(- sj?i)} ds1 ..-. ds,, fos ds, J;- (p(71) (k, klZ.) X I1J'=j expf - Y-.A i(,) Ik,. 110(sj- +)}d **s

fJ ds, * * *1IJH= exp{- EmflEAJ(kj) IkmI2/sj} dsi ds,,

and

n-l(i ) nI Ik77/ ) x exp{- >7nEAj($7) Iknl 21S} I

H J mA(z kn, ILimEAj(7z,) Jkm1 213 j=l inzEAj(,7) 7neAj(F)7(

It is elementary to check that, due to 'pij <1,

(31) I 'Pi,7 a (kl, . 1 . <,k )|'1

Using Lemma 2, we infer that the left-hand side of (26) equals

(32) Jo I Lin E kin, ( Y; Ikm P12>1(Y) Q( )

(32) xE

x E[v i airl2Jl (o,dkn)] -LmEA, (7) U {n?+1}

Here the summation extends over all multiindices i = (ij, ..., Iin+?) such that

i1 = i, in+1 = j, all Feynman diagrams Y e AS and all proper functions o- of

order n. Using an elementary inequality stating that

1 _ -x/ C

? 25 + X

for a certain constant C independent of 8, x, we conclude that the absolute value of (32) is less than or equal to

-216 K K ~Pn-1, 8(a) 53(kr - kIn,i)dkmndkm, (33) 2t,n8+82 ZnE () l'

Jo J ?26+ EMnEA ,(b9) kn 72i`in I1#a-

with

n-I L~MEA.(,7) km

j1 828 = IMEAEA k2in

Suppose first that 2,8 > 1. There exists then a constant C, depending only

on n, /8 and K such that, for any mj c Aj(Y),

(34) LinE Aj(y) kin 8? /: + km

828 + EmEAj($7) kin 828 + kmj


So,

p 8'61,8 213?CHkin. (35) P1-i < - 88/1 2

828 + Ln,E A,z (Y) km j-1 828 + kn 2 + kIn,

for any choice of ni j E Aj(Y). Let m: j if j is not the right endpoint of an edge of the diagram Y. Otherwise, let mj be the closest free vertex to the left of j.

Consequently, the expression on the right-hand side of (35) is less than or equal to

(36) C H 81-fkm Hl (8/1 km +)I-5 I,l81

(mmn'EE(Y) 8 +km, mEA,,(Y)U{n+1} (8 +

where E(Y) denotes the set of the edges of the diagram Y and q7n > 0 is the number of left endpoints between a free vertex m E A1n(Y) and the next free vertex m' E A,7(Y), also by a convention q,,+? := -1. Note that

(37) 7 +2 , qm = n-2, mnEA,,(9)U{n+1}

with c7 denoting the cardinality of the set An(F). Applying (36) to (33), we deduce that

( fK (8/P + k)dk ) (n-cr, )/2 1-0,71L (]o (28?k2P8)k2al)

(38) K (k +- 8113)2+q??+q,, - 7 X dk

m (k2P + 828)2+q?,n+q?, k2a-1.

Here the summation extends over all Feynman diagrams Y from AS({l, .I. , n}) and all complete diagrams Y' made of the vertices of A,,{5') U {n + 1} and rm, m := (m,5m,l +?m, m,2n Using the definition of ( [see (7)], it is elementary to observe that

IK (88/P + k) dk

(8o28 ? k2/3)k2a-l ( Y

and

JK (k + 86/13)2+q,+qn,,m, dk c +

o (k2? + 82)2+?q+qnZ, k2al - V

with

3 - 2a - 2,8 (39)

(40) 4-2a -4/3 + (q?n

+ q7l,)(1-228)-r1 z, in,

0ny(mm) a? + 2B-1


x~~~

21

FIG. 2. Regions A, B and C for estimating .

Hence we obtain

(41) cnf < C +1 28(1 + ?(n-c-)y

with

2r(2 - a - 2,3) + L' [(1 - 2/)(q72 + q7n') rnz 7n] K.-

a + 23 - 1

Here the summation ' extends over the edges mm' of the diagram Y' for which y(mm') < 0 and

(42) r < (c1, + 1)/2

denotes the number of such edges. In case there are no such edges we set K := 0. The estimate of the right-hand side of (41) consists of considering all possible situations depending on signs of the expressions 3 - 2a - 2,3, 2 - a - 2,3 appearing in (39) and (40). As shown in Figure 2, there are three cases corresponding to three regions (A, B and C) in the (a, ,3) plane. In each region we can deduce that

(43) 1-07Y_1 < Cs

Indeed, in region A we have

4(1 - a - /3) + (1 - 2/)(n - c,,) K> 2(a + 2/-1)


and /j: (2(a + /3) - 1)/2(a + 2/3 - 1); in region B we have

-2a + cn(3 - 2a - 2/3) + (1 - 2/3)n K> 2(a + 23 - 1)

and u (2(a +3) - 1)/2 (a + 2/3 - 1); in region C we have K as in the previous case and /u := 1/(a + 2/3 - 1).

When, on the other hand, 2/3 < 1, we conclude that P<z-i 8(Y) < C for a certain constant C > 0. From (33) we obtain that

Ct8~~128 K

82 1 5m'~(km - km,)dkrndkm 7,, , < ct,,n+1-28 ,J ..J. Ek I

Ct,1nl-2802(l-a-8)8/1 Jo (Kk213 1d)k 2a- t -

In conclusion, we deduce that all terms _n vanish as s 1 0 when n > 2. Estimates of 2N. Note that according to (24)

N = 2? N+2 ds ds... J E[ESN 1WN(S1, ..., SN+1, ?X(SN+1))

? V(O, 0)] dSN+l1

By the Cauchy-Schwarz inequality we get that

I2N 12 < 4t2%4(1-8)+2NE V(0, 0)2

x max 2EJ ... ESS>S?> ESNl1 (45) 0<S_t182 / >Sj> ...>SN+1 >0

2

X WN(Sl, ..., SN, SN+?1, X(SN+1)) ds. dsN+1

The stationarity of the Lagrangian velocity field implies that the maximum in (45) is equal to

max E fSds'/ f EOWN(sl.,sN,0,O)dsl ...dsN2 O<s<t/825 >s'>s .. >SN>0

(46) < C max E Jds' EOVWN1(SI, ,SN-1,SN, ) O<s<t/,-25 0>Sj> ...>SN >0

2

x dsl... dSN EIV(O,0)12.

In the last line we used that WN E HN(V) and the resulting hypercontractive property of LP-norms with respect to a Gaussian measure on the space HN(V) (cf. Proposition 1). Using subsequently Lemma 2 to represent the conditional


expectations on the right-hand side of (46), we deduce that the left-hand side of the preceding formula is less than or equal to

C -,xEE i (k, kN)NSQy (47) 2

x Hl Vij,,Ul,,,(0, dk7n) 77mEAN(y)

with some lI j, < 1. The summation in (47) above extends over all Feynman diagrams Y E Se, the relevant proper functions o- and multiindices i.

Thus, we have

(48) w2 < Ct4,,2N+4(1l28) [K K 'y 73(k1 kmn)dkm'dkin N L 1oPN, ()i k2a-l mm' ~m

Here the summation extends over all possible diagrams YS E AS,1({l, ..., N}), ' E Sc(AN () U N + AN(Y)). The product is taken over all edges of 5'

with AN(y) denoting the set of free edges of Y. Let CN be the cardinality of AN(Y). Arguing as in (38), when 2/3 > 1, we obtain that, for q,, > 0, m E AN(y) U N + AN(y7) as in (36) satisfying

2CN?+2yqm = 2N

we have

2 (j~~~~K (e8/138 -1 k) dk N-cN | 12 < Ct4?,2N+4(1-25) E(/ 6+#pk- MN ~~~~~~~(828 ? k2j8)k2a-l (49) K k( h /J3 2+q,f+l?q, dk

x H (k2,8 +825 X kal

nim,

The ranges of the sum and the product in (49) remain the same as in (48). Repeating the argument made after (42), we deduce that there exists /Lj > 0 such that

(50) IN 2 < Ct48NI-i-88

The same inequality, with /Lj = 1, holds also when 2,3 < 1. This can be deduced repeating the corresponding argument used to obtain (44). We infer there- fore that 2N vanishes as 8 4 0 for a sufficiently large N. In conclusion, we proved that the left-hand side of (22) tends to Dt2H as 8 4 0, provided that a +,3 > 1. 0

REMARK. The foregoing argument can be used to infer, via an application of the hypercontractivity property of the LP-norms over Gaussian measures on HN(V), that for an arbitrary p > 1 and T > 0 there exists a constant C > 0 such that, for any T > t > s > 0 8 > O,

(51) E x,(t) - x(s) I P< C(t - s)2HP.


PROOF OF LEMMA 2. We prove the lemma by induction. The case n = 1 is obvious by choosing p(iO) =)1. Suppose that the result holds for n. For the sake of convenience we assume without loss of generality that sn 2 = 0. Then

(52) E0 WV1+1, (S1, ., s711, X)

= Eo {V(sn+, X) VE,,I+ 1W71,i(si, Sn, x)}.

By virtue of the inductive assumption we can represent ESi+ W1Z using (28) and as a result (52) becomes

E Eo [J | (Pi ,07(kl, . , k7l)

(53) x expj- L Iki2I+(sn- sn?)}Pn 1(Y)Q(5z)

XVO(Sn+ln X, dknz+1 Vt v Vi ,7771 (sn+1D Xn dkin)] mEA,,(Y)

To calculate (53), we decompose each V., i(s, x, dk) as

(54) V,J i(s, x, dk) = V, js, x, dk) + V1 i(s, x, dk)

and use (13)-(12) where

(55) V (s, x, dk) e- Ik V (txdk)

is the orthogonal projection of V on YK t. Expression (53) becomes

LE0[ ... ( (kl,... k)

(56)

M EA,,(5/7)

with

K(5y) L= E Vgl+1(s, x, dk,,+1)

jEAII(7)U{fn+1}

Vtr n QveZlZ ins QIII x VjI Vc (s, x, dk1n)


The term corresponding to g 1 vanishes, as is clear from the following calculation:

E jf. I a )(kl, k.. , - 1 (Y) Q(y)

xVOl(s, x, dk,,+,) Vt I1 VCIZ i ... (s, x, dkm))

(57) A,

= V E J JD(n(kj, ... , k7l)P71-1(Y)Q(P)

x Vo(s,x,dkn?i) mIkl I V i(S X,dk1n) =? M EA, (,7)

by homogeneity of the velocity field. By (13)-(12),

VO(s) x) dk,+, V vuln, i m (s x dkm)}

(58) M EA,,(,7)

- L kmm .Vo(s, x, dkn) x H1 VC (S, X, dkm) M' EA,j Y) M EA,,(Y)

where

(59) in' = | r7Z7, if m'-m, m 0t (T7n, otherwise.

By (55), (54), (58) and definition (29), (56) further reduces to

E E E E E (F7d km',il

i7+= n'EA,, (,9')' YinEA,, (Y) lkm I

x expt- E I knz 11sz+1 Pn(Y)Q(Y) mEA(Y')

(60) X fH V-T1Z,' i,l (t, X, dkm) inEA(6 ')

x H [1 -e-(Ik p2 + lkq120)(S_t)]

PqEE(,7')


with &, n =Oand -u+i,7Z' = o-TY and all incomplete Feynman diagrams 7' based on the set A,,(7) U {n + 1}. Lemma 2 follows with

(71?1) (n) k1? i,+ 'Pi, ..': (k, , kni) 4(i, 0(ki , k1 ) m T~~~~~~~~~LM'EA,J(Y) Ii'

x e1 [1- (Ip1kp 2+lkkqI20)Si+i] p^qEE(Y')

4. Proof of weak convergence. It is a straightforward matter to verify that the Gaussian processes

t/25

(61) y8(t) 81 V(s, O)ds, t > 0O

converge weakly to the fractional Brownian motion BH(t), t > 0, given by (8). In addition, we have

limsupE ly(t) P < +0o ?|0

for any p > 1, t > 0. We now prove that

lim E [Xs,i,(t1)

- Xs,i1(t2)IP' ' [Xs, im(tm) - X, iM(tM+l)]1

= E [BH, i1(tl) - BH, il(t2)] [BH,iM(tM)-BH,iM(tM+1)IPM

Equation (62) implies that the limiting law of the family of processes x8(t), t > 0, whose tightness, as 8 4 0, has been established in the previous section, is that of the fractional Brownian motion BH(t), t > 0. Equation (62) is a consequence of

(63) lin ? ilt1) -x8, i1(t2)]P . [x8 iM(tM) - Xe, iAI(tM?l)IP

- [Y8, il (tl) Y8, i, (t2)] P

[Y8, iM (tM) - Ye, iM (tM+l)PM O,

with y8(t) = (y8 1(t), . Y-, d(t)). Equation (63) follows from the next lemma.

LEMMA 3. For any positive integers M, Pl, ..., PM, multiindices i1 c{1,...,d}PJ for j= ,...,Mand tl > > tM > tM+l =0, wehave

(64) lim E[Z i, (t2, tl) ... Z?, iM (tM+l, tM)

-W? i (t2, tl) ... W, iM (tM+1, tM)] 0.


Here for any integer N > 1, multiindex i = (i,..., iN) E {1, d}N and t > s, we define

Z$ji (SI t) HN1 * 171 Vj (sP' 8x(SP))dsj ... dSN

AN(S, t)

and N

W?i (S, t) 'EN 1 H V(Sp,O) dsi ... dsN, AN(S, t)

=

with AN(S, t) = {(Sl, *, SN) t/?8 > Sl > * > SN > S/?8}

PROOF. To avoid cumbersome expressions that may obscure the essence of the proof, we consider only the special case of M = 1 and t1 = t, t2 = 0. The general case follows from exactly the same argument. We shall proceed with the induction argument on Pi = P. The case when P = 1 is trivial because the stationarity of the relevant processes implies that the expression under the limit in (64) vanishes. In fact, as a consequence of the remark made after the proof of Lemma 2, we can conclude that, for any q> 1,

lim sup E Zs,?(O, t) <7 +.

Assume now that (64) has been proven for a certain P - 1 > 1 and that for any q > 1 we have

(65) lim sup E Z? i (0, t) <)+o.

In analogy with (23) we write that

N-1

(66) EZ? i(O, t) - L X9(O, t) + IN(70 t), 11=0

with

--g?n At)1 := 1 (t) E{S2Wi(1 'X(S2))

(67) P

X H V (Sp, ex(SP))} ds( d 2 ... dsp, p=2

titNOn t = P+N+1l .. EN E{SN WiN S(N ) I?X(Sl1 N+1))

(68) x H Vi (sp,ex(sp))ds'(N'dS2 dsp.

p=2


Here

A\p (s, t) := I(s(" , S",**. Sp) : t/ 2 > Sl 2 2> >S S?}

wih1l): (Si, 17 Si, n+1) We say that t > s, > s, where s =(sl, s , S7) iS

an ordered n-tuple, that s> > s, when t > s, > S,, > s. The argument used in the proof of Lemma 2 together with (65) shows that

lim 7 (, t) = 0

for n > 1 and

l?iM 'wN (0, t) = O,

provided that N is chosen sufficiently large. Asymptotically then, as 8 4 0 the behavior of EZ(P)(O, t) is the same as that of the term

-90 (0, t) :=? |***|(0) Et Vil (sj, Ex(S2))

(69) p x H Vp(sp,8x(sp))dSl... dsp.

p=2

In order to deal with (69), we need a generalization of the argument used in the proof of Lemma 2. Let us introduce some additional notation. For any multiindex i = (ij, ..., ip) and p > 1 we define Wf"'1 by induction as follows. We set

i l i (s1,. . ., Sp, X)

:= Vi,(Si, x) Vip(SP, x) - El Vj(sl, x) .. Vi (Sp, x)}

and assuming that Wip17 (Sl, .Sp-, s(n)7I x) has been defined for any

ordered n + 1-tuple s (1) = (sp,i **^ Sp,77?i) < Sp-, we set

WiPl i... 1 * * *li Sp-1, Sp?), X)

VWpn?1) X VS X := V i i, (Sj,***, p ,,

X * (pn+2, X

for any ordered n + 2-tuple s = (sp 1' **., Sp 7+ Sp 7 Expanding the left-hand side of (69) in analogy with (23), we obtain

'x 0 )

=

|

.

IA(o)(O' t) E IVi, (Sl, --X(S2)) Vi2 (S2, --X(S2))l

x E H V p(Sp, ?X(Sp))} ds1 dS2... dsp p=3

N-1

+ E -L1, ni(O t) + 1 N(O t), 77-n


where

J-gi,n(0 t) = f f(1,,) EjES3Wl iD2(5lr s2 , ?X(S3))

(70) p X H V (sp, ?x(sp))J dsi ds l) ds2... dsp,

p=3

P+N+1 ... E E w 2, N Sl dS2N) E sX .dp

1, NO ( t7) | A(1, N) (0 t) E S2, N+I 'I l2 72 (S1 X 2, N+1))

(71) x H Vi(sp, ?x(sp)) Js,sl d N) ds3 dsp, p=3

A (,n

(0, t) = l |s1 (n)

: 3 p t/? 2 > 51 > S

(n > . > Sp, >l.

We represent the conditional expectations appearing in (70) and (71) using a generalization (Lemma 4) of Lemma 2.

To formulate it, we need a generalized notion of a proper function, which we call a p-proper function. Let p be a positive integer. The p-proper function of order 1 is unique and is given by o-(i) = 0, i = 1,., p. Any p-proper function, o-', of order n + 1 is generated from a p-proper function o- of order n as follows. For some q < p + n,

O'(p + n + 1) 0,

(72) O'(k) :o(k) for k < n + p and k 0 q,

ur'(q) 1 -(q).

We also distinguish a special class of Feynman diagram pfs(B) a diagram Y of order n+p belongs to pSs(B) if Ak(Q9) is not empty for all k p, ..., n+p.

LEMMA 4. For any positive integer p, s, > ... sp > s(p-) > s, a multiindex i (ij, ..., ip) E {1, ..., d}P, we have

ES WFI'7 (sl, . ,Sp_l, s (n-1) I X)

(73) x exp{- EA lkm 12,(Spn-S)Pp , -

x m l V j," .(Js x, dkm), MeA7+P(Y)


where 9(p"') are functions satisfying I (pv7")I < 1 and

n+p-1

PP,itQJ) lkml j=p mEAj(17)

(74) x expj- E lkm12P(p J ppj - sP j_p+1)j, meA(,7)

Q( ) H E[Vi7, -71,; (O, dkn] )V i7n X 7;z, (O, dkm')] mm' eE(7)

The summation is over all multiindices j = (jl,..., jJ7?+p) such that jlp 1i all 5 E p4S and all p-proper functions o- of order n. Here by a convention Sp, o := SP-.

The proof of Lemma 4 is exactly the same as that of Lemma 2 and is omitted. Using Lemma 4 and the argument presented in the foregoing to demon-

strate that .Y0(O, t) is asymptotically equal to EZ(P)(O, t), as 8 4 0, we can show that

P+1~~~~~~~ ? | . . . | E( Vi1 (s 1, EX (S3 ) V i2 (52, -v ?( S3))

p

X Hl Vi (sp EX(sp)) dsids2 dsp p=3

is asymptotically equal to EZ(?i)(O, t), as 8 4 0. Repeating the preceding argument p times, we obtain (64). Finally, we notice that the hypercontractivity properties of the LP norms over Gaussian measure space allow us to conclude that (65) holds with P - 1 replaced by P. L

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DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA DAVIS, CALIFORNIA 95616-8633 E-MAIL: [email protected]

INSTITUTE OF MATHEMATICS MARIA CURIE-SKLODOWSKA UNIVERSITY LUBLIN POLAND

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